Subgroups — #89610

Order 448.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1131	C₂	1	2	yes	maximal, complemented
2	C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₇	C₂²	1	2	yes	—
8	C₂×C₇×C₈	C₂²	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	(C₂×D₇)⋊C₄	C₄	1	2	yes	—
11	C₄⋊Dic₇	C₄	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	#1131	—	2	4	no	—
15	#436	—	7	1	no	maximal, Sylow
16	C₂²×D₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	C₂×C₄	1	2	yes	—
21	D₂₈	—	2	2	no	—
22	C₄×D₇	—	2	2	no	—
23	C₄×D₇	—	2	2	no	—
24	C₇×C₈	—	2	2	no	—
25	C₇⋊C₈	—	2	1	no	—
26	C₇⋊C₈	D₄	1	2	yes	complemented
27	C₂×M₁₆	—	7	2	no	—
28	C₄×D₄	—	7	1	no	—
29	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
30	C₂²⋊C₈	—	7	2	no	—
31	C₄×C₈	—	7	1	no	—
32	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₇	—	2	2	no	—
34	C₂×D₇	—	2	2	no	—
35	C₂×D₇	—	2	2	no	—
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	M₁₆	1	2	yes	complemented
38	C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	#41	1	1	yes	characteristic
40	Dic₇	—	2	2	no	—
41	C₂×D₄	—	7	1	no	—
42	C₂²×C₄	—	7	2	no	—
43	M₁₆	—	14	4	no	—
44	C₂×C₈	—	7	1	no	—
45	C₂×C₈	—	7	1	no	—
46	C₂×C₈	C₂×D₇	1	2	yes	—
47	C₄⋊C₄	—	7	1	no	—
48	C₂²⋊C₄	—	7	2	no	—
49	C₄²	C₂×D₇	1	1	yes	characteristic
50	C₂×C₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₇	#131	1	1	yes	characteristic
53	D₇	—	4	2	no	—
54	C₂³	—	7	2	no	—
55	D₄	—	14	2	no	—
56	C₂×C₄	—	7	2	no	—
57	C₂×C₄	—	14	2	no	—
58	C₂×C₄	—	14	2	no	—
59	C₂×C₄	C₄×D₇	1	1	yes	characteristic
60	C₂×C₄	C₄×D₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	7	2	no	—
64	C₈	—	14	1	no	—
65	C₇	#436	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	14	2	no	—
67	C₂²	—	14	2	no	—
68	C₂²	—	14	2	no	—
69	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	14	2	no	—
72	C₄	#1131	1	2	yes	complemented
73	C₄	D₄×D₇	1	1	yes	characteristic
74	C₄	#1161	1	1	yes	characteristic
75	C₂	—	28	2	no	—
76	C₂	#6456	1	1	yes	characteristic
77	C₂	C₂×#1131	1	1	yes	characteristic
78	C₂	#6470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₇×C₄⋊C₈	C₂
prime radical	C₂×D₂₈	C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#436
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#1131	C₂	semidirect
3. C₇×C₄⋊C₈	C₂	semidirect
4. (C₂×D₇)⋊C₈	C₂	semidirect
5. C₄×C₇⋊C₈	C₂	semidirect
26. C₇⋊C₈	D₄	semidirect
29. C₄⋊C₈	D₇	semidirect
37. C₄×C₇	M₁₆	semidirect
65. C₇	#436	semidirect
72. C₄	#1131	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#1131	2
C₄×D₂₈	2
C₇×C₄⋊C₈	2
(C₂×D₇)⋊C₈	2
C₄×C₇⋊C₈	2
#436	7